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140       XIII    INTEGRATION

(1 3.9.1 3) In order that a mapping/ : X -* R should be integrable, it is necessary
and sufficient that f should be measurable and that the upper integral ^*(|/|)
be finite.

Necessity. Since |/| is integrable (13.7.4), clearly ju*(|/|) is finite. Next
we remark that, for each integer n, there exists by hypothesis a function/, e >
such that/rg/, and X/ -/)  V (13.7.1). Replacing/, by inf / (12.7.5),

we may assume that the sequence (/,) is decreasing. If g is the limit of this
sequence (equal to its lower envelope), we have f^g and ^(gf) =
lim n(fn f)  0 (13.8.1). In other words, /is almost everywhere equal to

n-xx)

the limit of the sequence (/,). Now all functions belonging to J are measurable
(1.3.9.11), and another application of Egoroff's theorem (13.9.10) then proves
that /is measurable.

Sufficiency. We may assume that/is finite, by replacing/by an equivalent
function (13.6.4). By (13.9.12) there exists a sequence^) of measurable step
functions with compact supports such that \gn\^ \f\ and the sequence (gn)
converges to /almost everywhere. Now gn is a linear combination of charac-
teristic functions of relatively compact measurable sets ; since such sets are
integrable (13.9.2), it follows that gn is integrable. Hence, applying (13.8.4)
again, /is integrable.

(13.9.14) (i) If f is an integrable function and A is a measurable set, then
the function fq>^ is integrable.

(ii) If(An) is an increasing sequence of integrable sets, whose union is the
complement of a negligible set N, then

\fdfjL = lim \f<pAndn.

J                -*> J

We may assume without loss of generality that / is everywhere finite
(13.6.4). Then/(pA is measurable (13.9.6), and we have

from which (i) follows by (13.9.13). The assertion (ii) follows from the
dominated convergence theorem (13.8.4).

In the situation of (13.9.14), we put j fyA dp = JA /<//*, or JA/(*) dfi(x);
this number is called the integral off over A. Also (if A is measurable) we
write J * fdu in place of J *fq>A dp.ned from semi-continuous